Magic squares on Abelian groups

Abstract

Let $(\hat{I}“,+)$ be an Abelian group of order $n^2$ and MS${}_{\hat{I}}“(n)$ be an $n\times n$ array whose entries are all elements of $\hat{I}“$. Then MS${}_{\hat{I}}“(n)$ is a $\hat{I}“$-magic square if all row, column, main and backward main diagonal sums are equal to the same element $\hat{I}¼\in \hat{I}“$. We prove that for every Abelian group $\hat{I}“$ of order $n^2$, $n>2$, there exists a magic square MS${}_{\hat{I}}“(n)$ where the square entries are elements of $\hat{I}“$.